- #1

Wendel

- 10

- 0

## Homework Statement

I'm trying to understand the intuition behind path-connectedness and simple-connectedness in finite topological spaces. Is there a general methodology or algorithm for finding out whether a given finite topological space is path-connected?

## Homework Equations

how can I determine which of the following topologies are path-connected?

for the three-element set:

1. {∅,{a,b,c}}

2. {∅,{c},{a,b,c}}

3. {∅,{a,b},{a,b,c}}

4. {∅,{c},{a,b},{a,b,c}}

5. {∅,{c},{b,c},{a,b,c}}

6. {∅,{c},{a,c},{b,c},{a,b,c}}

7. {∅,{a},{b},{a,b},{a,b,c}}

8. {∅,{b},{c},{a,b},{b,c},{a,b,c}}

I won't list them out, but for the four-point set there are 33 inequivalent topologies. One of which is the pseudocircle X={a,b,c,d} which has the topology {{a,b,c,d},{a,b,c},{a,b,d},{a,b},{a},{b},∅}.

## The Attempt at a Solution

The discrete topology is totally disconnected. Any map from the unit interval to the indiscrete topology is continuous, so it must be path-connected. Furthermore the particular point topology is path-connected. The pseudocircle is clearly path-connected since the continuous image of a path-connected space is path-connected. Furthermore it is not simply connected. From Wikipedia, connectedness and path-connectedness are the same for finite topological spaces.

Thank you, apologies for the long post.[/B]